Unimodular Lattices
A unimodular lattice is a discrete subset of n dimensional Euclidean
space with the property that the space is equal to its dual, where two
two vectors (representing points in the space) are orthogonal if
their inner product is an integer.
This definition can also be extended to other spaces with the
appropriate choice of integers.
To learn about the connection between self-dual codes and unimodular lattices, see
the excellent book, "Sphere Packings, Lattices and Groups" by J.H. Conway
and
N.J.A. Sloane and the references therein.
To learn about the relationship between unimodular lattices and codes over
Z_{2^m} see the following papers:
- Type II Self-Dual Codes over Finite Rings and Even
Unimodular Lattices, Journal of Algebraic Combinatorics, Volume 9, Issue 3,
1999, p. 233-250, with T.A. Gulliver and M. Harada.
-
Type II Codes, Even Unimodular Lattices, and Invariant Rings,
IEEE-IT, Volume 45, Number 4, 1999, 1194-1205,
with E. Bannai, M. Harada, and M. Oura.
- Shadow Lattice and Shadow Codes,
Discrete Math, Vol. 219 (2000), 49-64,
(with M. Harada and P. Sole.)
- Splitting the Shadow , with A. Bonnecaze,
Y. Choie and P. Sole.
- Generalized Shadows of Codes over Rings .
To learn more about the relationship between codes and complex
unimodual lattices check out the following papers:
- Type II Codes Over F_2 + u F_2,
IEEE-IT, vol 45. No. 1, January 1999,
with M. Harada, P. Gaborit, and P. Sole.
- Codes over Z_4 + w Z_4,
Complex Lattices and Hermitian Modular Forms,
with YoungJu Choie.
- Codes over F_3 + v F_3, with R. Chapman, P. Sole,
and P. Gaborit.
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